Optimal. Leaf size=10 \[ \cosh (x)-2 \log (\cosh (x)+1) \]
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Rubi [A] time = 0.04, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2667, 43} \[ \cosh (x)-2 \log (\cosh (x)+1) \]
Antiderivative was successfully verified.
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Rule 43
Rule 2667
Rubi steps
\begin {align*} \int \frac {\sinh ^3(x)}{(1+\cosh (x))^2} \, dx &=-\operatorname {Subst}\left (\int \frac {1-x}{1+x} \, dx,x,\cosh (x)\right )\\ &=-\operatorname {Subst}\left (\int \left (-1+\frac {2}{1+x}\right ) \, dx,x,\cosh (x)\right )\\ &=\cosh (x)-2 \log (1+\cosh (x))\\ \end {align*}
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Mathematica [A] time = 0.02, size = 13, normalized size = 1.30 \[ \cosh (x)-4 \log \left (\cosh \left (\frac {x}{2}\right )\right )-1 \]
Antiderivative was successfully verified.
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fricas [B] time = 2.70, size = 48, normalized size = 4.80 \[ \frac {4 \, x \cosh \relax (x) + \cosh \relax (x)^{2} - 8 \, {\left (\cosh \relax (x) + \sinh \relax (x)\right )} \log \left (\cosh \relax (x) + \sinh \relax (x) + 1\right ) + 2 \, {\left (2 \, x + \cosh \relax (x)\right )} \sinh \relax (x) + \sinh \relax (x)^{2} + 1}{2 \, {\left (\cosh \relax (x) + \sinh \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.12, size = 21, normalized size = 2.10 \[ 2 \, x + \frac {1}{2} \, e^{\left (-x\right )} + \frac {1}{2} \, e^{x} - 4 \, \log \left (e^{x} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 11, normalized size = 1.10 \[ \cosh \relax (x )-2 \ln \left (1+\cosh \relax (x )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.30, size = 23, normalized size = 2.30 \[ -2 \, x + \frac {1}{2} \, e^{\left (-x\right )} + \frac {1}{2} \, e^{x} - 4 \, \log \left (e^{\left (-x\right )} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.94, size = 10, normalized size = 1.00 \[ \mathrm {cosh}\relax (x)-2\,\ln \left (\mathrm {cosh}\relax (x)+1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.52, size = 58, normalized size = 5.80 \[ - \frac {2 \log {\left (\cosh {\relax (x )} + 1 \right )} \cosh {\relax (x )}}{\cosh {\relax (x )} + 1} - \frac {2 \log {\left (\cosh {\relax (x )} + 1 \right )}}{\cosh {\relax (x )} + 1} - \frac {\sinh ^{2}{\relax (x )}}{\cosh {\relax (x )} + 1} + \frac {2 \cosh ^{2}{\relax (x )}}{\cosh {\relax (x )} + 1} - \frac {2}{\cosh {\relax (x )} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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